Neumann equation

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

The main problem is to calculate (/ (q, H-r)/(q, t- -r)) of Eq. (2). To achieve this goal, one first considers E(r,f) as a well-defined, deterministic quantity. Its effect on the system may then be determined by treating the von Neumann equation for the density matrix p(f) by perturbation theory the laser perturbation is supposed to be sufficiently small to permit a perturbation expansion. Once p(i) has been calculated, the quantity... 

Ignoring relaxation, the spin evolution is governed by the Liouville-von-Neumann equation... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

By analogy the propagation of a density matrix, which corresponds to the solution of the Liouville-von Neumann equation 231... 

Dependent Liouville-von Neumann Equation Dissipative Evolution. 

Another important hierarchy of equations is obtained by applying the (MCM) to the matrix representation of the Liouville-von Neumann Equation LVNE) [12,13]. In this way the p-order Contracted Liouville-von Neumann Elquation (p-CLVNE) is obtained [4]. It will be shown here that The structure of a particular p-CSE, that involves the higher order CSE s for a given state, can be replaced by an equivalent set of equations, 1-CSE and 1-CLVNE, but for the whole spectrum, i. e. involving all the states. 

Let us start the statistical theory with the von Neumann equation for the density matrix... 

Exercise. When the combined Hamilton operator of two systems is additive, H = Hl +H2, the Neumann equation (1.20) has solutions of the form (1.22). The two systems are independent at all times. 

The evolution of the molecule is described by the generalized Liouville-von Neumann equation [35, 36]... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

In an NMR experiment, the energy of the lattice is practically constant (the lattice has a large heat capacity). It is therefore assumed that the lattice is always in a state of thermodynamic equilibrium. Thus, it is possible to use a semi-classical description of its interactions with the spin system. Within this approach, the Liouville-von Neumann equation of motion for a spin system is given by ... 

To describe consistently cotunneling, level broadening and higher-order (in tunneling) processes, more sophisticated methods to calculate the reduced density matrix were developed, based on the Liouville - von Neumann equation [186-193] or real-time diagrammatic technique [194-201]. Different approaches were reviewed recently in Ref. [202]. 

We consider also the other method, based on the equations of motion for operators. From Liouville - von Neumann equation we find (all c-operators are Heisenberg operators in the formula below, (t) is omitted for shortness)... 

Having established the Hamiltonian, the next step is to derive an equation of motion for the density matrix. One can start with the Liouville-von Neumann equation for the complete system-plus-bath density matrix a(t)... 

Starting point for the derivation of the Nakajima-Zwanzig identity is the Lioville-von Neumann equation for the density operator a of the complete system, i.e. relevant system plus environment. As stated before h = 1 is used and thus the QME reads... 

So within a few steps it is possible to derive the Nakajima-Zwanzig identity starting from the Lioville-von Neumann equation. 

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